Numbers. Principles Of Digital Design. Number Representations

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1 Prncples Of Dgtal Desgn Numbers Number Representatons Decmal, Bnary Number System Complement Number System Fxed Pont and Floatng Pont Numbers

2 Postonal Number System Each number s represented by a strng of dgts, n whch the poston of each dgt has an assocated weght = In general, any decmal number D of the form d m 1 d m 2 d 1 d 0.d 1 d 2 d n has the value Radx Least sgnfcant dgt Radx pont Most sgnfcant dgt m 1 D = d m 1 10 m d d d n 10 n = = n d 10 Copyrght by Danel D. Gajsk 2! EECS31/CSE31, Unversty of Calforna, Irvne

3 Bnary Number System General form of a bnary number: b m 1 b m 2 b 1 b 0.b 1 b 2 b n Its value s equvalent to Examples B = m 1 = n b 2 Least sgnfcant bt (LSB) Bnary pont Most sgnfcant bt (MSB) Radx = = = = = = = = Copyrght by Danel D. Gajsk 3! EECS31/CSE31, Unversty of Calforna, Irvne

4 Converson from Decmal to Bnary m 1 D = d 10 = (( ((d m 1 )10 + d m 2 )10 + )10 + d 1 )10 + d 0 = 0 n 1 B = b 2 = (( ((b n 1 )2 + b n 2 )2 + )2 + b 1 )2 + b 0 = 0 Dvdng the top equaton by 2, we obtan the quotent S and remander R S = ( ((b n 1 )2 + b n 2 )2 + )2 + b 1 R = b 0 Start S = D = 0 Dvde S by 2 S = quotent b = remander S = 0? yes Contnung dvdng S by 2, we obtan the new quotent S and next bnary dgt no = + 1 Done Copyrght by Danel D. Gajsk 4! EECS31/CSE31, Unversty of Calforna, Irvne

5 Decmal-to-Bnary Example Problem: Convert 179 to bnary, Soluton: B = Start S = D = = 89 remander 1 (b 0 ) 89 2 = 44 remander 1 (b 1 ) 44 2 = 22 remander 0 (b 2 ) 22 2 = 11 remander 0 (b 3 ) 11 2 = 5 remander 1 (b 4 ) 5 2 = 2 remander 1 (b 5 ) 2 2 = 1 remander 0 (b 6 ) 1 2 = 0 remander 1 (b 7 ) Dvde S by 2 S = quotent b = remander S = 0? yes Therefore, = b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0 = no = + 1 Done Copyrght by Danel D. Gajsk 5! EECS31/CSE31, Unversty of Calforna, Irvne

6 Converson from Bnary to Decmal n 1 b 2 = 0 Problem: Convert to decmal, B = = (( ((b n 1 )2 + b n 2 )2 + )2 + b 1 )2 + b 0 Soluton: D = 179, Start D 0 x = 1 1 x = 2 2 x = 5 5 x = x = x = x = x = 179 Therefore, b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0 = => 179 b D = 0 = n 1 D = D x 2 + b yes = 0 no = 1 Done Copyrght by Danel D. Gajsk 6! EECS31/CSE31, Unversty of Calforna, Irvne

7 Sgn Magntude Representaton A sgn magntude number <s, m>, conssts of two parts: Sgn (s) and Magntude (m) The sgn s ether + or The magntude s an nteger between 0 and the largest representable value Examples: = = Copyrght by Danel D. Gajsk 7! EECS31/CSE31, Unversty of Calforna, Irvne

8 Two s Complement Number System m 1 Two s-complement of a number B = (ex. 0110) s equal to: m m B = 2 B = (( 2 1 ) B ) + 1 ( ) Proof: If dgt complement b = (2 1) b = 1 - b (2 m 1) B = ( (2 1) (2 1) (2 1) (b m 1 b m 2 b 0 ) ) then = ( (2 1) b m 1 ) ( (2 1) b m 2 ) ( (2 1) b 0 ) m 1 = b m 1 b m 2 b 0 = = B (1001) = 0 Therefore, B = B ' + 1 (1010 = ) where B s a negatve number of B, snce B + B = 0 ( = 10000) b ' = 0 b 2 Copyrght by Danel D. Gajsk 8! EECS31/CSE31, Unversty of Calforna, Irvne

9 Complement Number System Decmal Two s Complement Sgn- Magntude or Two s Complement and Sgn-Magntude Representatons Copyrght by Danel D. Gajsk 9! EECS31/CSE31, Unversty of Calforna, Irvne

10 Floatng-pont Numbers Floatng-pont numbers have the form mantssa (radx) exponent Snce radx s mplct, only mantssa and exponent must be represented explctly Floatng-pont numbers are fxed-pont numbers gven by the mantssa, whose radx pont s specfed by the exponent Exponent s represented n the excess-code format called characterstc, obtaned by addng a bas to the exponent: bas = (radx s /2 ) 1 where s s equal to the number of bts n the exponent feld Mantssa sgn Sgned exponent Mantssa magntude General format Sgn Excess-127 characterstc Normalzed Fracton Sgn Excess-1023 characterstc Normalzed Fracton 32-bt standard Impled bnary pont 64-bt standard Impled bnary pont Copyrght by Danel D. Gajsk 10! EECS31/CSE31, Unversty of Calforna, Irvne

11 Fxed-pont vs. Floatng-pont The range s the nterval of numbers from the largest to the smallest representable number The precson s the amount of numbers n a number nterval 4-dgt fxed number 4-dgt floatng-pont number Integer Mantssa Exponent Representable numbers Range ~ 10 4 ~ Precson ~ 100 ~ 1 Example 1001 numbers between 1000 and , 1001, 1999, numbers between 1000 and x10 2, 11x10 2, 19x10 2,20x10 2 Copyrght by Danel D. Gajsk 11! EECS31/CSE31, Unversty of Calforna, Irvne

12 Summary Dgts and Numbers Decmal Bnary Number Representaton Sgn- magntude Twos-Complement Floatng-pont Copyrght by Danel D. Gajsk! EECS31/CSE31, Unversty of Calforna, Irvne

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